MATH 295 PROBLEMS 5

IGOR KRIZ

1. Compute:

lim_{n→∞} \frac{n^{2}}{10 + n\sqrt{n}}.

2. Prove that for every set S of real numbers there exists an interval I such that we have

S ⊆ I,

and every interval J with S ⊆ J satisfies I ⊆ J. [For two sets X, Y we write X ⊆ Y if every element of X is also an element of Y.]

3. For what choices of numbers a, b, c and d will the function

f(x) = \frac{ax + b}{cx + d}

satisfy f(f(x)) = x for all x for which the left hand side is defined?

4. Find the maximal possible domain of the function defined by the formula

f(x) = \sqrt{1 − \sqrt{1 − x^{2}}}.

[This means: for what values of x ∈ ℝ is the formula on the right hand side defined?]

5. Prove that for any real numbers a_{1}, ..., a_{n}, b_{1}, ..., b_{n} we have

a_{1}b_{1} + ... + a_{n}b_{n} ≤ \sqrt{a_{1}^{2} + ... + a_{n}^{2}}\sqrt{b_{1}^{2} + ... + b_{n}^{2}}.

[Hint: square the expression, and simplify.]